How to Convert from Other Number Bases to Decimal System?

Questions on conversion of numbers in some base to some other base is very common in competitive examination. Here in this post I present a simple technique to help you do such conversions.

First, let us understand what do we mean by number bases or systems. In our decimal number system, the rightmost position represents the “ones” column, the next position represents the “tens” column, the next position represents “hundreds”, etc. Therefore, the number 123 represents 1 hundred and 2 tens and 3 ones, whereas the number 321 represents 3 hundreds and 2 tens and 1 one.

The values of each position correspond to powers of the base of the number system. So for our decimal number system, which uses base 10, the place values correspond to powers of 10:
… 1000 100 10 1

… 103 102 101 100

Binary to Decimal Conversion

Other number systems use different bases. The binary number system uses base 2, so the place values of the digits of a binary number correspond to powers of 2. For example, the value of the binary number 10011 is determined by computing the place value of each of the digits of the number:

1 0 0 1 1 the binary number

24 23 22 21 20 place values

So the binary number 10011 represents the value

= (1 x 24) + (0 x 23) + (0 x 22) + (1 x 21) + (1 x 20)

= 16 + 0 + 0 + 2 + 1

= 19

Conversion from Other Number Bases to Decimals

The same principle applies to any number base. For example, the number 2132 base 5 corresponds to